SYSTEMS OF MATING 7 



p ' of the mating AA x AA is k 2 by the product rule of probability. The probability 

 r of the matings AA x Aa or Aa x ^ is £ -2/ plus 2/ -£ , or 4£ / , by the product and 

 addition rules of probability, and so forth. In the table, P(AA x Aa) is the probability 

 of A A x Aa and its reciprocal Aa x AA. 



The relative frequencies of the three genotypes in G ls obtained by adding the last 

 three columns of table 2, are: 



?(AA) = k x = (k + l ) 2 



P(Aa) = 21, = 2(k + / )(/ + m ) = 2x y , 



P(aa) = m x = (/ + w ) 2 = y 2 . 



If random mating ensues, that is, if the probability of any mating type is the 

 product of the genotypic probabilities of the mates, the mating types of G x and the 

 genotypes of their progeny in G 2 can be represented as functions of the initial allelic 

 frequencies, x and y , as in table 3. The genotypic frequencies in G 2 obtained by 

 adding the last three colums of table 3 are : 



P(AA) =k 2 = V(V + 2*0*0 + y 2 ) = a-o 2 , 

 P(Aa) = 2/ 2 = 2x Q y (x 2 + 2x Q y + y 2 ) = 2x y , 

 P{aa) = m 2 = y 2 (x 2 + 2x y + y 2 ) = y 2 . 



Table 3 



MATING-TYPE FREQUENCIES OF Gi AND THE GENOTYPE FREQUENCIES OF THEIR PROGENY 



Genotypes in G 2 

 Mating types in Gi AA Aa aa 



■ P(AA x AA) = p[ = k x 2 = V .v 4 



P(aa x aa) — p'[ = m, 2 = y 4 y 4 



P(AA x aa) — q, — 2k 1 m 1 — 2x 2 y 2 2x 2 y 2 



P(AA x Aa) = r x = 4kJ, = 4x 3 y 2x 3 y 2x 3 y 



P(aa x Aa) — s, = Al x mi — 4x y 3 — 2x y 3 2x y 3 



P(Aa x Aa) = v, = \l x 2 = 4x W *oW 2x 2 y 2 x 2 y 2 



The mating-type frequencies of G x and the genotype frequencies of G 2 are functions of the allele 

 frequencies of G in the random-mating system. 



These are identical with the genotypic frequencies of G x . Hence the mating-type 

 frequencies of G 2 will also be identical with the mating-type frequencies of G x . It 

 follows that p' n + 1 = p[; . . . ; v n + 1 = v x . After one generation of random mating, the 

 probabilities of the six kinds of matings remain constant for any number of generations 

 of random mating. Random mating thus preserves the genetic variability of the popu- 

 lation. The restriction — "after one generation" — may be removed if the initial geno- 

 typic frequencies, k , 2/ , m are related to the allelic frequencies, * and y , as 



/- — X 2 



2/ = 2x y , 



